Q.Integrate the following function: $(ax + b)^2$

The key idea is to expand the square and then integrate term-by-term using the Power Rule.

First, expand the square:

Now integrate each term separately:

Apply the Power Rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$: …

We are asked to integrate $(ax + b)^2$ with respect to $x$. The natural instinct might be to try a substitution like $u = ax + b$, but that would require an extra factor of $1/a$ from the chain rule. However, since the expression is a simple binomial squared, expanding it is far more straightforward and avoids any risk of sign errors.

The Power Rule for integration says that for any real number $n \neq -1$,

This rule works term-by-term when we have a sum of powers of $x$, which is exactly what we get after expanding.

Let’s go step by step.

1. Expand the square.

   Recall that $(ax + b)^2 = a^2 x^2 + 2ab x + b^2$.

   This is just the algebraic identity $(p+q)^2 = p^2 + 2pq + q^2$, with $p = ax$ and $q = b$.

2. Rewrite the integral as a sum.

3. Apply the Power Rule to each term separately.

   • For $a^2 x^2$: $\int a^2 x^2 \, dx = a^2 \cdot \frac{x^{3}}{3} = \frac{a^2 x^3}{3}$.
   • For $2ab x$: $\int 2ab x \, dx = 2ab \cdot \frac{x^{2}}{2} = ab x^2$.
   • For $b^2$: $\int b^2 \, dx = b^2 x$ (since $x^0$ integrates to $x^1/1$).

4. Combine the results and add the constant of integration.

   $\frac{a^2 x^3}{3} + ab x^2 + b^2 x + C.$ …

Method: Expand a binomial power before integrating

A binomial like $(ax+b)^2$ can be integrated by expanding or by a linear substitution; expanding is safest when you need the result in polynomial form.

Steps

Step 1: Expand.

$(ax+b)^2=a^2x^2+2abx+b^2$, using $(p+q)^2=p^2+2pq+q^2$.

Step 2: Integrate term by term with the power rule.

Apply $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}$ to each term. …

Common Mistakes

Mistake 1: Writing $\frac{(ax+b)^3}{3}$ without the $\frac1a$.

Why it's wrong: by the chain rule the substitution answer is $\frac{(ax+b)^3}{3a}$; omitting $\frac1a$ is off by a factor of $a$. Correct approach: divide by $a$, or expand and integrate term by term.

Mistake 2: Mis-expanding the middle term. …